Resonances of a potential well with a thick barrier (Q2862275)

The long-time behavior of the solutions of the wave equation in the case of a potential well surrounded by a thick barrier can be determined from the resonances, i. e. the eigenvalues, of the eigenvalue problem NEWLINE\[NEWLINE-\psi ''+V(x)\psi =k^2 \psi ,\;\psi '+ik\psi =0 \text{ for }x=L,\;\psi '-ik\psi =0 \text{ for }x=-L. \tag{1}NEWLINE\]NEWLINE It is assumed that there is \(a<L\) such that \(V(x)=V_0\) for \(a\leq |x|\leq L\) and that \(|V(x)|\leq V_0\) for \(|x|\leq a\); \(L-a\) is the thickness of the barrier. The eigenvalues have negative imaginary parts, and it is assumed that they are simple. The authors investigate the behavior of the spectrum as \(L\to \infty \), in which case the limiting problem has bound and unbound states. In Section 3 it is shown that near bound states there are approximate eigenvalues which are exponentially small in \(L\). The large eigenvalues \(k_m\) have the asymptotic distribution Re\,\(k_m\sim\pi\frac mL\) and Im\,\(k_m\sim-\frac1L\log\frac mL\). A numerical example is given for the case that the potential inside the well is zero. Also, for high frequency resonances, numerical results are obtained.